Let $(X,\Sigma,\mu)$ be a finite measure space and let $L^2(\mu,\mathbb{R}^d)$ denote the Bochner space of strongly measurable functions taking values in $\mathbb{R}^d$.  Let 
$$
\begin{aligned}
T &:L^2(\mu,\mathbb{R}^d)\rightarrow L^2(\mu,\mathbb{R})\\
& f\mapsto \|f\|_2,
\end{aligned}
$$
where $\|\cdot\|_2$ is the Euclidean norm on $\mathbb{R}^d$. Fix a $\phi\in\Gamma_0(L^2(\mu,\mathbb{R}))$.  

Is there a known expression for
 $
\operatorname{Prox}_{T\circ \phi}
$
in terms of $\operatorname{Prox}_{\phi}$, $\phi$, and $T$?